12.5 Free-Surface Flows
385
with overturning free surfaces, but the gas enclosed by the liquid phase will
not feel buoyancy effects and will therefore behave in an unrealistic manner.
Kawamura and Miyata (1994) used Eq. 12.29) to calculate the distribution
of the density function (the product of the density and the VOF) and to
locate the free surface, which is the contour with c = 0.5. The computation
of the motion of the liquid and gas flows are done separately. The free surface
is treated as a boundary at which the kinematic and dynamic boundary
conditions are applied. Cells which become irregular due to being cut by
the free surface require special treatment (variable values are extrapolated
to nodal locations lying on the other side of the interface). The method was
used to calculate flows around ships and submerged bodies.
Alternatively, one can treat both fluids as a single fluid whose properties
vary in space according to the volume fraction of each phase, i.e.:
p = p l c + p a ( l - c ) ,
P = P I C + P ~ ( ~ - C ) ,
(12.30)
where subscripts 1 and 2 denote the two fluids (e.g. liquid and gas). In this
case, the interface is not treated as a boundary so no boundary conditions
need to be prescribed on it. The interface is simply the location where the fluid
properties change abruptly. However, solution of Eq. (12.29) implies that the
kinematic condition is satisfied, and the dynamic condition is also implicitly
taken into account. If surface tension is significant at the free surface, it can
be taken into account by treating the force as a body force. Methods of this
kind were presented by Brackbill et al. (1992), Lafaurie et al. (1994), Ubbink
(1997), and Muzaferija and PeriC (1999), among others. These methods can
also deal with merging and fragmentation in multiphase flows.
The surface-tension force acts only in the region of interface, i.e. in partially filled cells, since in full or empty cells the gradient of c is zero:
However, there are problems when the surface-tension effects become dominant, like in the case of droplets or bubbles whose diameter is of the order
of 1 mm or less and which move with very low velocity. In this case, there
are two very large terms in the momentum equations (the pressure term and
the body force representing the surface-tension effects) which have to balance each other; they are the only non-zero terms if the bubble or droplet is
stationary. Due to the fact that curvature of the interface also depends on c,
grad c
K = -div (---) ,
l grad cl
it is difficult to ensure on an arbitrary 3D grid that the two terms are identical,
so their difference may cause the so called parasitic currents. These can be
avoided by using special discretization methods in 2D (see Scardovelli and
Zaleski, 1999, for some examples of such special methods); we do not know at
385
with overturning free surfaces, but the gas enclosed by the liquid phase will
not feel buoyancy effects and will therefore behave in an unrealistic manner.
Kawamura and Miyata (1994) used Eq. 12.29) to calculate the distribution
of the density function (the product of the density and the VOF) and to
locate the free surface, which is the contour with c = 0.5. The computation
of the motion of the liquid and gas flows are done separately. The free surface
is treated as a boundary at which the kinematic and dynamic boundary
conditions are applied. Cells which become irregular due to being cut by
the free surface require special treatment (variable values are extrapolated
to nodal locations lying on the other side of the interface). The method was
used to calculate flows around ships and submerged bodies.
Alternatively, one can treat both fluids as a single fluid whose properties
vary in space according to the volume fraction of each phase, i.e.:
p = p l c + p a ( l - c ) ,
P = P I C + P ~ ( ~ - C ) ,
(12.30)
where subscripts 1 and 2 denote the two fluids (e.g. liquid and gas). In this
case, the interface is not treated as a boundary so no boundary conditions
need to be prescribed on it. The interface is simply the location where the fluid
properties change abruptly. However, solution of Eq. (12.29) implies that the
kinematic condition is satisfied, and the dynamic condition is also implicitly
taken into account. If surface tension is significant at the free surface, it can
be taken into account by treating the force as a body force. Methods of this
kind were presented by Brackbill et al. (1992), Lafaurie et al. (1994), Ubbink
(1997), and Muzaferija and PeriC (1999), among others. These methods can
also deal with merging and fragmentation in multiphase flows.
The surface-tension force acts only in the region of interface, i.e. in partially filled cells, since in full or empty cells the gradient of c is zero:
However, there are problems when the surface-tension effects become dominant, like in the case of droplets or bubbles whose diameter is of the order
of 1 mm or less and which move with very low velocity. In this case, there
are two very large terms in the momentum equations (the pressure term and
the body force representing the surface-tension effects) which have to balance each other; they are the only non-zero terms if the bubble or droplet is
stationary. Due to the fact that curvature of the interface also depends on c,
grad c
K = -div (---) ,
l grad cl
it is difficult to ensure on an arbitrary 3D grid that the two terms are identical,
so their difference may cause the so called parasitic currents. These can be
avoided by using special discretization methods in 2D (see Scardovelli and
Zaleski, 1999, for some examples of such special methods); we do not know at